1 \documentclass[12pt]{article}
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2 \title{Strong Separation}
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4 \date{\vspace{-2em}\today{}}
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11 \section*{The Calculus}
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12 \subparagraph{Syntax}
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13 \[\begin{array}{lll}
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14 \tm, \tmtwo & \ddef & \var \mid \tm\,\tmtwo \mid \Lam\var{\tm\Comma\vec\tm} \\
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15 n & \ddef & \Lam\var{n\Comma\vec n} \mid \var\,\vec n \\
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17 C & \ddef & \Box \mid C\,\tm \mid \tm\,C \\
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18 P & \ddef & \vec\tm \Comma \Box \Comma \vec\tm \mid \vec\tm \Comma C[\Lam\var P] \Comma \vec\tm \\
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21 \subparagraph{Reduction rules}
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22 \[\begin{array}{lll}
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23 P[C[(\Lam\var{\tm\Comma\vec\tm})\,\tmtwo]] & \Red{}{\var\in \tm,\vec\tm} &
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24 P[C[\tm\Subst\var\tmtwo]\Comma \vec\tm\Subst\var\tmtwo]\\
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25 P[C[(\Lam\var{\tm\Comma\vec\tm})\,\tmtwo]] & \Red{}{\var\not\in\tm,\vec\tm} &
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26 P[C[\tm] \Comma \vec\tm\Comma\tmtwo]\\
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29 \subparagraph{Properties}
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31 \item Every term is normalizing iff it is strongly normalizing.
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32 \item Ogni strategia e' perpetua!
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37 \section*{Separation}
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39 \newcommand{\PathEmpty}{\epsilon}
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40 \newcommand{\PathAbs}[1]{\mathtt{abs}(#1)}
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41 \newcommand{\PathArg}[3]{\mathtt{arg}_{#2}^{#1}(#3)}
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42 \newcommand{\PathHd}{\mathtt{hd}}
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44 \newcommand{\GarbageOf}[1]{\operatorname{Garb}(#1)}
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45 \newcommand{\HeadOf}[1]{\operatorname{head}(#1)}
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46 \newcommand{\FstOf}[1]{\operatorname{fst}(#1)}
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47 \newcommand{\DegOf}[1]{\operatorname{deg}(#1)}
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48 \newcommand{\SubtmOf}[2]{#1\preceq #2}
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49 \newcommand{\OfHead}[2]{#1_{{\mid}#2}}
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50 \newcommand{\SubtmsOf}[1]{\operatorname{Sub}(#1)}
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51 \newcommand{\Div}{\mathtt{d}}
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52 \newcommand{\Conv}{\mathtt{c}}
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53 \newcommand{\Const}{\mathtt{K}}
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54 \newcommand{\NamedBoundVar}[1]{\texttt{bvar(}#1\texttt{)}}
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55 \newcommand{\AC}[1]{{\color{violet}#1}}
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57 % \item \textbf{$\boldsymbol\sigma$-separation.}
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58 % \textcolor{red}{come definirlo? con le variabili? con i termini?
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59 % problematico in cbv}
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60 % Two terms are $\sigma$-separable iff there exists a substitution
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61 % $\sigma$ such that \textcolor{red}{???}
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62 % \item \textbf{Semi-$\boldsymbol\sigma$-separation.}
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63 % Two terms are semi-$\sigma$-separable iff there exists a substitution
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64 % $\sigma$ such that -- in short -- it makes one diverge and the other one converge.
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65 % \item \textbf{Our subproblem:} Semi-$\sigma$-separating two (usual) $\boldsymbol\lambda$-terms
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66 % (in deep normal form)
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67 \item \textbf{Subterm:} $\SubtmOf\tm\tmtwo$ means that $\tm$ is an ($\eta/\Omega$)-subterm of $\tmtwo$.
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68 \item \textbf{Subterm at position $\boldsymbol\pi$:} TODO
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69 \item $\boldsymbol\sim_{\boldsymbol\pi}$
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70 \item \textbf{Distinction:} Let $\var\defeq \HeadOf D$. Let $T_{\var} \defeq \{\tm \preceq T \mid \HeadOf{\tm} = \var \}$.
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71 $C_{\var}$ is $D$--\emph{distinct} iff it is empty, or there exists path $\pi$ s.t.:
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73 \item \emph{effective} for $D$, cioe' $\FstOf{\pi} \leq \DegOf{D}$;
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74 \item $\forall \tm\in D_\var$, $\tm_\pi \neq \Omega$;
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75 \item $\exists \tm\in C_\var$ s.t. $\tm \not\sim_\pi D$;
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76 \item $\{\tm\in C_\var \mid \tm \sim_\pi D\}$ is $D$--distinct.
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80 \item $\bullet$ $\bullet$ $\bullet$ $\bullet$ $\bullet$
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81 $\bullet$ $\bullet$ $\bullet$ $\bullet$ $\bullet$
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82 $\bullet$ $\bullet$ $\bullet$ $\bullet$ $\bullet$
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83 \texttt{ prove di nuove definizioni di ac:}
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85 \item \textbf{Set of subterms:} %$\SubtmsOf{\tm} \defeq \{ \tmtwo \mid \SubtmOf\tmtwo\tm \}$
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87 \SubtmsOf{\var} & \defeq \{\var\} \\
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88 \SubtmsOf{\tm\,\tmtwo} & \defeq \SubtmsOf\tm \cup \SubtmsOf\tmtwo \cup \{\tm\,\tmtwo\} \\
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89 \SubtmsOf{\lambda \var.\, \vec\tm} & \defeq \{\tmtwo\{\Const/\var\} \mid \tmtwo \in\SubtmsOf{\vec\tm}\} \\
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91 \AC{Note: $\SubtmsOf\cdot$ replaces bound variables with $\Const$ when going under abstractions.}
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92 \item \textbf{Subterm at position:}
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94 \text{Paths: } \pi & ::= \PathEmpty \mid \PathHd \mid \PathArg i \var \pi \mid \PathAbs\pi
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97 Given a path, one can retrieve from a term (if possible) the subterm at that position.
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99 Since the path may go through abstractions, bound variables that become free
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100 are renamed to variables of the form $\NamedBoundVar\pi$
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101 (where $\pi$ is the path in the original inert leading to the abstraction binding that variable).
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103 % \newcommand{\GetSubtm}[2]{\operatorname{GetSubtm}(#1\texttt{;}#2)}
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104 \newcommand{\GetSubtm}[2]{{#1}_{#2}}
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105 \[\begin{array}{ll}
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106 \GetSubtm\tm\pi & \defeq \GetSubtm\tm{\underline\pi} \\
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107 \GetSubtm\tm{\rho[\underline\PathEmpty]} & \defeq \tm \\
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108 \GetSubtm{(x\,t_1\cdots t_n)}{\rho[\underline\PathHd]} & \defeq x \\
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109 \GetSubtm{(x\,t_1\cdots t_n)}{\rho[\underline{\PathArg i \var \pi}]} & \defeq
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110 \GetSubtm{(t_i)}{\rho[\PathArg i \var {\underline\pi}]} \mbox{(if } 1 \leq i\leq n \mbox{)} \\
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111 \GetSubtm{(\lambda x.\, t)}{\rho[\underline {\PathAbs \pi}]} & \defeq
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112 \GetSubtm\tm{\rho(\PathAbs{\underline\pi})}
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113 \{\var\mapsto\NamedBoundVar{\rho[\PathEmpty]}\} \\
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114 \GetSubtm{\tm}{\rho(\underline{\PathAbs \pi})} &
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115 \defeq \GetSubtm{(\lambda \var.\, \tm\,\var)}{\rho(\PathAbs {\underline\pi})} \text{ (with } x \text{ fresh) (eta)}\\
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116 % \Omega_-^- & \defeq \Omega \\
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118 \item \textbf{Head restriction:} $\OfHead T \var \defeq \{\tm \in T \mid \HeadOf{\tm} (\defeq \tm_{\PathHd}) = \var \}$
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119 \item \textbf{Telescopic garbage chain:} $\{\langle\tm_1,\pi_1\rangle,\ldots,\langle\tm_n,\pi_n\rangle\}$ is a $-$ if $\forall i$:
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120 \[\tm_{i+1} \in \SubtmsOf{\text{garbage of } \GetSubtm{(\tm_i)}{\pi_i}}\]
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121 \item \textbf{Distinction:} \underline{$S$ is $\{\langle\Div_1,\pi_1\rangle,\ldots,\langle\Div_n,\pi_n\rangle\}$--distinct} iff (one of the following three):
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123 \item $\OfHead S {\HeadOf \Div}$ is empty and $n=1$
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125 OR: let $\Div\defeq\Div_1$ in:
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127 \item there exists a path $\pi$ s.t.
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128 \item (Effective) $\pi$ is \emph{effective} for all $\Div_i$ s.t. $\HeadOf{\Div_i} = \HeadOf{\Div}$
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129 \item $\forall \tm\in \OfHead{\SubtmsOf{\Div_i}}{\HeadOf\Div}$, $\tm_\pi \neq \Omega$;
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130 \item (Useful) $\exists s\in \OfHead S{\HeadOf\Div}$ s.t. $s \not\sim_\pi \Div$;
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131 \item $S\setminus\{s\in \OfHead S{\HeadOf\Div} \mid s \not\sim_\pi \Div\}$ is $D$--distinct.
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135 \item $S'$ is $\{\langle\Div_2,\pi_2\rangle,\ldots,\langle\Div_n,\pi_n\rangle\}$--distinct, where:
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136 \[S' \defeq S \mathrel{\cup} \SubtmsOf{\{\text{garbage of } s \text{ along } \pi_1 \mid s\in \OfHead{S}{\HeadOf\Div}\}} \]
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139 \item \textbf{Semi-$\sigma$-separability: } $(\Div,\Conv)$ are semi-$\sigma$-separable
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140 IFF there is $\Div_1$ (an $\Omega$--approximation of a subterm of $\Div$ with
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141 at most one garbage, and without stuck variables)
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142 and a telescopic garbage chain $D\defeq\{\langle\Div_1,\pi_1\rangle,\ldots,\langle\Div_n,\pi_n\rangle\}$ s.t.
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143 $\SubtmsOf\Conv$ is $D$--distinct.
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146 \item $\bullet$ $\bullet$ $\bullet$ $\bullet$ $\bullet$
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147 $\bullet$ $\bullet$ $\bullet$ $\bullet$ $\bullet$
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148 $\bullet$ $\bullet$ $\bullet$ $\bullet$ $\bullet$
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150 \item \textbf{Unlockable variables.}
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151 We use the following contexts:
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152 $E ::= \Box \mid E\, t \mid t\, E \mid \Lam\var{\vec t\Comma E \Comma \vec t} $.
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153 A variable $\var$ is \emph{unlockable} in a context $E$ if:
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155 \item it is not bound in $E$, or
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156 \item $E[\cdot] = E'[\vartwo\, \vec\tm \, (\Lam{\cdots\var\cdots}{\vec\tmtwo \Comma E'[\cdot]\Comma\vec\tmthree})]$
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157 and $\vartwo$ is unlockable in $E'$.
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160 Transformation removing an unlockable variable bound at position $\pi$:
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161 \[\tau_{n::\pi}[\alpha] := \lambda x_1..x_n\,x.\, \alpha\,\vec x\,(\tau_\pi[x])\]
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163 % \textcolor{red}{For every term there exists an equivalent term with no unlockable variables}
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168 % \section*{NP-hardness}
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169 % \newcommand{\Pacman}{\Omega}
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170 % \newcommand{\sep}{\cdot}
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171 % \begin{example}[Graph 3-coloring]\label{example:3col}
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172 % Let $G=(V,E)$ be a graph, with $N \defeq |V|$.
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173 % We encode the problem of finding a 3-coloring of $G$ in the following problem of semi-$\sigma$-separation:
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174 % \[\begin{array}{cl}
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175 % \Uparrow & x \sep t_1^1\,t_1^2\,t_1^3 \sep t_2^1\,t_2^2\,t_2^3 \sep \cdots t_n^1\,t_n^2\,t_n^3 \\
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177 % \Downarrow & x \sep \Pacman\,\Pacman\,\Pacman \sep t_2^1\,t_2^2\,t_2^3 \sep \cdots t_n^1\,t_n^2\,t_n^3 \\
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178 % \Downarrow & x \sep t_1^1\,t_1^2\,t_1^3 \sep \Pacman\,\Pacman\,\Pacman \sep \cdots t_n^1\,t_n^2\,t_n^3 \\
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179 % \vdots & \vdots \\
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180 % \Downarrow & x \sep t_1^1\,t_1^2\,t_1^3 \sep t_2^1\,t_2^2\,t_2^3 \sep \cdots \Pacman\,\Pacman\,\Pacman \\
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183 % Where: $\dummy$ is (probably) a variable, $\bomb\defeq \lambda\_.\,\bot$, and the $a$'s are defined as follows:
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187 % \item $\begin{array}{ll}
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188 % a_1^1 \defeq & \lambda\_. \, x \sep y\bomb\bomb \sep\bomb\ldots \\
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189 % a_1^2 \defeq & \lambda\_. \, x \sep \bomb y\bomb \sep\bomb\ldots \\
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190 % a_1^3 \defeq & \lambda\_. \, x \sep \bomb\bomb y \sep\bomb\ldots \\
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193 % \item $a_2^1 \defeq \begin{cases}
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194 % \lambda\_.\, x \sep \bomb \dummy\dummy \sep y\bomb\bomb \cdot \bomb\ldots & \text{if } (v_1, v_2)\in E \\
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195 % \lambda\_.\, x \sep \dummy\dummy\dummy \sep y\bomb\bomb \cdot \bomb\ldots & \text{if } (v_1, v_2)\not\in E \\
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202 % \begin{definition}[Index notation]
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203 % Let $t = x \sep x_1^1 x_2^1 x_3^1 \sep x_1^2 x_2^2 x_3^2 \sep \ldots \sep x_1^m x_2^m x_3^m$. Then: \[t[\,_k^j] \defeq x_k^j.\]
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206 % Let $z_0$, $z_1$, $z_2$ be variables.
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209 % \[a_k^j[\,_{k'}^{j'}]\defeq\begin{cases}
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210 % \bomb & \text{if } j<j' \\
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212 % \bomb & \text{if } k\neq k' \\
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213 % y & \text{if } k = k' \\
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214 % \end{cases} & \text{if } j = j' \\
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216 % \bomb & \text{if } k = k' \\
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217 % \dummy & \text{if } k \neq k' \\
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218 % \end{cases} & \text{if } (v_j,v_{j'}) \in E \\
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219 % \dummy & \text{if } (v_j,v_{j'}) \not\in E
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222 % Attenzione! Le $a$ vanno protette da lambda ($\lambda\_$)!
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224 % % Dimensione del problema: circa $(3\times m^2)^2$.
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226 % Intuitively, if $\sigma(x)$ ``uses'' $a_j^i$, then $\sigma$ colors $v_j$ with color $i$.
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228 % \begin{lemma}[Extraction of the coloring]
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229 % Let $\sigma$ be a substitution which is a solution for the semi-separation problem. Then for example:
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232 % \item $\operatorname{color}(v_1) = 2$ iff
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233 % \[(x \sep \Pacman\,\bomb\,\Pacman \sep \bomb\,\bomb\,\bomb \,\sep \bomb\,\bomb\,\bomb \,\sep \cdots \sep \bomb\,\bomb\,\bomb)\,\sigma \to \bot\]
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234 % \item $\operatorname{color}(v_2) = 3$ iff:
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235 % \[(x \sep \dummy\,\dummy\,\dummy \sep \Pacman\,\Pacman\,\bomb \,\sep \bomb\,\bomb\,\bomb \,\sep \cdots \sep \bomb\,\bomb\,\bomb)\,\sigma \to \bot\]
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239 % % Where $\Pacman \equiv \lambda\_.\,\Pacman$.
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243 \section[Tentativi X-separability]{Tentativi $\mathbf X$--separability (July, 15$^{\mathbf{th}}\div\infty$)}
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244 \newcommand{\perm}{\pi}
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245 \newcommand{\Perm}[2]{\pi[#1,#2]}
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246 \newcommand{\xK}{\kappa}
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247 \newcommand{\xN}{n}
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248 \newcommand{\LAM}[2]{\Lambda_{#2,#1}}
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249 \newcommand{\LAMNK}{\LAM\xN\xK}
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250 % \newcommand{\kn}{$(\kappa,n)$}
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251 \newcommand{\knnf}{$\xK{}$-nf}
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252 \newcommand{\Lams}[1]{\operatorname{lams}(#1)}
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253 \newcommand{\Args}[1]{\operatorname{args}(#1)}
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254 \newcommand{\Head}[1]{\operatorname{head}(#1)}
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255 \newcommand{\nf}[1]{#1{\downarrow}}
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256 \newcommand{\Nat}{\mathbb{N}}
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257 \begin{definition}[$\Lams\cdot,\Args\cdot,\Head\cdot$]
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258 \[\Lams{\lambda\vec\var.\,\vartwo\,\vec\tm} \defeq |\vec\var| \]
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259 \[\Args{\lambda\vec\var.\,\vartwo\,\vec\tm} \defeq |\vec\tm| \]
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260 \[\Head{\lambda\vec\var.\,\vartwo\,\vec\tm} \defeq \vartwo \]
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263 \begin{definition}[$\xK$-normal forms]
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264 First recall that terms in normal form have the shape
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265 $\lambda\vec\var.\,\vartwo\,\vec\tm$, where the terms $\vec\tm$ are in normal form too.
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267 We define inductively the set of \knnf s (where $\xK$ is a natural number):
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268 $\lambda\vec\var.\,\vartwo\,\vec\tm$ is a \knnf{} iff
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269 $|\vec\var|\leq\xK$%
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270 %, $|\vec\tm|\leq\xN$
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271 , and every term in $\vec\tm$ is a \knnf{} as well.
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274 \begin{definition}[Normal form $\nf\cdot$]
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278 \begin{definition}[Permutator $\Perm\cdot\cdot$]~
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279 \[\Perm i j \defeq \lambda\vec\var\vartwo.\,\vartwo\, \vec\alpha\,\vec\var\,\vartwo\]
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280 where $\vec\var$, $\vec\alpha$ and $\vartwo$ are fresh variables,
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281 with $|\vec\var| = i$ and $|\vec\alpha| = j$.
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284 {\color{yellow}\begin{definition}[Good permutation]
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286 $\Perm i j$ is \emph{good} for $\tm$ if
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287 $i<\Args\tm$ and $j \geq \xK + i + 1$.
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292 % Let $\vec\alpha$ fresh variables.
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293 % If $|\vec\alpha|\geq \lams\tm$,
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294 % then the normal form of $\tm\,\vec\alpha$ is inert.
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297 \begin{lemma}[Monotonicity]\label{l:k-nf-mono}
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298 If $\tm$ is a \knnf{}, then it is also a $\xK'$-nf
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299 for every $\xK' \geq \xK{}$.
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303 Let $\tm$ a \knnf, $\var$ a variable, $i$ a natural number, and $\xK'\defeq \xK + i + 1$.
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304 Then for every $j \geq \xK'$, $\nf{\tm\Subst\var{\Perm i j}}$ is defined and it is a $\xK'$-nf.
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307 By (course-of-value) structural induction on $\tm$.
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308 Let $\tm=\lambda\vec\vartwo.\,\varthree\,\vec\tmtwo$ and $\sigma\defeq\Subst\var{\Perm i j}$.
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311 \item Case $\varthree\neq\var$: $\nf{(\lambda\vec\vartwo.\,\varthree\,\vec\tmtwo)\sigma} = \lambda\vec\vartwo.\,\varthree\,\vec{(\nf{\tmtwo\sigma})}$.
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312 By \ih{} each term in $\vec\tmtwo\sigma$ is a $\xK'$-nf.
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313 We conclude since by hypothesis $|\vec\vartwo|\leq\xK{}<\xK'$.
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315 \item Case $\varthree=\var$ and $i<|\vec\tmtwo|$:
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316 $\tm\sigma \to^* \lambda\vec\vartwo.\, (\tmtwo_i\sigma)\vec\alpha \vec{(\tmtwo\sigma)} $.
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317 Now, by \ih{} $\nf{\tmtwo_i\sigma}$ is a $\xK'$-nf, and since $|\vec\alpha|\geq\xK'$,
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318 $\nf{(\tmtwo_i\sigma\vec\alpha)}$ is inert (\textcolor{red}{Serve lemma?}).
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319 Therefore $\nf\tm = \lambda\vec\vartwo.\,\nf{(\tmtwo_i\sigma\vec\alpha)} \nf{\vec{(\tmtwo\sigma)}}$,
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320 and again by \ih{} it is also a $\xK'$-nf.
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321 \item Case $\varthree=\var$ and $|\vec\tmtwo|\leq i$:
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322 $\tm\sigma \to^* \lambda\vec\vartwo\vec\varthree.\, \varthree\vec\alpha \vec{(\tmtwo\sigma)}\vec\varthree $ for some $\varthree\in\vec\varthree$
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323 and $|\vec\varthree| = i + 1 - |\vec\tmtwo|$.
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324 Conclude by \ih{} and because $|\vec\vartwo\vec\varthree| \leq \xK{} + i + 1 - |\vec\tmtwo| \leq \xK{} + i + 1 = \xK'$.
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330 % Let $\tm$ a \knnf, and $\Head\tm=\var$.
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331 % If $i<\Args\tm$ and $j \geq \xK + i + 1$,
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332 % then $\tmtwo\defeq\nf{\tm\Subst\var{\Perm i j}}$ is defined;
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333 % $\Lams\tm=\Lams{\tmtwo}$; $\tmtwo$ is a $(j,\xN + j)$-nf.
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335 % \begin{proof}\color{red}\TODO{}
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336 % By induction on the normal form structure of $\tm$:
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337 % let $\tm=\lambda\vec\var.\,\vartwo\,\vec\tmtwo$.
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338 % By \ih{}, $\nf{\tmtwo\Subst\var\perm}$ are $(j,\xN+j)$-nfs.
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340 % \item if $\vartwo=\var$, then
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341 % $\nf{\tm\Subst\var\perm} = \lambda\var_1\ldots\var_{???}.\,\var\,\vec\tmthree$
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342 % \item if $\vartwo\neq\var$, then $\nf{\tm\Subst\var\perm} = \lambda\vec\var.\,\vartwo\,\vec\tmthree$
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343 % where $\vec\tmthree\defeq \nf{\vec\tmtwo\Subst\var\perm}$.
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344 % Conclude by inductive hypothesis.
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349 For every \knnf{s} $\tm,\tmtwo$,
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350 every fresh variable $\var$,
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351 $\perm\defeq\Perm{i}{i+k+1}$ permutator,
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352 $\tm\EtaEq\tmtwo$ iff $\nf{\tm\Subst\var\perm}\EtaEq\nf{\tmtwo\Subst\var\perm}$.
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355 The implication from left to right is trivial.
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356 We now prove that if $\nf{\tm\Subst\var\perm}\EtaEq\nf{\tmtwo\Subst\var\perm}$
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357 then also $\tm\EtaEq\tmtwo$.
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